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Intuitionistic first-order logic extended with a restricted form of Markov'sprinciple is constructive and admits a Curry-Howard correspondence, as shown byHerbelin. We provide a simpler proof of that result and then we studyintuitionistic first-order logic extended with unrestricted Markov's principle.Starting from classical natural deduction, we restrict the excluded middle andwe obtain a natural deduction system and a parallel Curry-Howard isomorphismfor the logic. We show that proof terms for existentially quantified formulasreduce to a list of individual terms representing all possible witnesses. Ascorollary, we derive that the logic is Herbrand constructive: whenever itproves any existential formula, it proves also an Herbrand disjunction for theformula. Finally, using the techniques just introduced, we also provide a newcomputational interpretation of Arithmetic with Markov's principle.

On Natural Deduction for Herbrand Constructive Logics II: Curry-Howard Correspondence for Markov's Principle in First-Order Logic and Arithmetic